thermally induced mechanical adelaide young response of

Tizian Bucher1

Advanced Manufacturing Laboratory,

Department of Mechanical Engineering,

Columbia University,

New York, NY 10027

e-mail: [email protected]

Adelaide YoungAdvanced Manufacturing Laboratory,



New York, NY 10027


Min ZhangMem. ASME

Laser Processing Research Center,

School of Mechanical and

Electrical Engineering,

Soochow University,

Suzhou 215021, Jiangsu, China


Chang Jun ChenMem. ASME

Laser Processing Research Center,

School of Mechanical and

Electrical Engineering,

Soochow University,

Suzhou 215021, Jiangsu, China


Y. Lawrence YaoFellow ASME

Advanced Manufacturing Laboratory,



New York, NY 10027


Thermally Induced MechanicalResponse of Metal Foam DuringLaser FormingTo date, metal foam products have rarely made it past the prototype stage. The reason isthat few methods exist to manufacture metal foam into the shapes required in engineeringapplications. Laser forming is currently the only method with a high geometrical flexibil-ity that is able to shape arbitrarily sized parts. However, the process is still poorly under-stood when used on metal foam, and many issues regarding the foam’s mechanicalresponse have not yet been addressed. In this study, the mechanical behavior of metalfoam during laser forming was characterized by measuring its strain response via digitalimage correlation (DIC). The resulting data were used to verify whether the temperaturegradient mechanism (TGM), well established in solid sheet metal forming, is valid formetal foam, as has always been assumed without experimental proof. Additionally, thebehavior of metal foam at large bending angles was studied, and the impact of laser-induced imperfections on its mechanical performance was investigated. The mechanicalresponse was numerically simulated using models with different levels of geometricalapproximation. It was shown that bending is primarily caused by compression-inducedshortening, achieved via cell crushing near the laser irradiated surface. Since this mech-anism differs from the traditional TGM, where bending is caused by plastic compressivestrains near the laser irradiated surface, a modified temperature gradient mechanism(MTGM) was proposed. The densification occurring in MTGM locally alters the materialproperties of the metal foam, limiting the maximum achievable bending angle, withoutsignificantly impacting its mechanical performance. [DOI: 10.1115/1.4038995]

Keywords: laser forming, metal foam, mechanical response, digital image correlation,bending mechanism

1 Introduction

Since its introduction in the first half of the 20th century, metalfoam has been the subject of countless research studies. Manyresearchers have demonstrated the great potential of the materialdue to its high strength-to-weight ratio and its outstanding noiseand shock absorption capacity [1,2]. Its material properties havebeen studied in great detail [3,4], and numerous potential indus-trial applications have been identified [5,6].

Despite all these efforts, metal foam products have rarely madeit past the prototype stage. The reason is that industrial applica-tions often call for intricately shaped parts that are challenging tomanufacture. Near-net-shape manufacturing methods for metalfoam do exist, such as three-dimensional printing [7], as well as apowder metallurgy process [8]. The former, however, is a notori-ously slow process, limited to small production volumes and partsizes. The latter, in turn, requires molds, limiting it to large pro-duction volumes and moderately sized parts.

Metal foam is normally manufactured in generic shapes, suchas slabs or sheets, and subsequently bent to the required shape.

This approach is less costly, can be implemented on a much largerscale, and yields a more uniform density distribution. The chal-lenge associated with this method is that metal foam is prohibi-tively difficult to bend due to the foam’s high moment of area andbending stiffness. Conventional mechanical bending methodshave shown no success to date. 3-point bending, for instance, gen-erates high tensile stresses that exceed the foam failure strengthand cause immediate failure [9]. Similarly, hydroforming wasshown to cause excessive densification [10]. Therefore, an alterna-tive bending method is needed that does not exert mechanicalforces.

Laser forming has been investigated as an alternative bendingmethod, since it is a noncontact method that does not require part-specific molds. The process is considered an economic option forlow to mid production volumes because it can be used to formarbitrarily sized parts into a wide range of geometries without amold. Laser forming has been studied extensively for sheet metalforming, and many aspects have been investigated, such as strainrate effects [11] and process synthesis considerations [12].

Within the last decade, several research groups have attemptedlaser forming of metal foam and reported positive results. Experi-ments have been conducted with open-cell foams [13], closed-cellfoams [14], and closed-cell foams with outside skin [15], eachcase showing that bending angles up to 45 deg are feasible. The

1Corresponding author.Manuscript received March 31, 2017; final manuscript received January 4, 2018;

published online February 12, 2018. Assoc. Editor: Hongqiang Chen.

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reported experimental work involves parametrical studies, inwhich bending angles have been measured for a range of differentprocessing parameters, such as power, spot size, scan speed, andcooling methods. Moreover, the process has been studied for arange of metal foam properties, such as densities, pore sizes, andsheet thicknesses. Additionally, issues related to microstructure[16] and heat treatments [17] have been addressed, and some pre-dictive capabilities have been developed [14,18,19].

From the aforementioned work, a rather complete pictureemerges about the processing window that is required for metalfoam laser forming, and some rudimentary numerical capabilitieshave been developed. However, the existing work has three essen-tial shortcomings.

First, none of the previous studies have addressed the underly-ing bending mechanism in sufficient detail. So far, it has beenassumed, without experimental proof, that the temperature gradi-ent mechanism (TGM), identified by Vollertsen [20] for sheetmetal laser forming, is always the governing bending mechanismin metal foam laser forming. While this assumption has beenexperimentally confirmed from a thermal standpoint [21], metalfoam does not meet the requirements of TGM from a mechanicalstandpoint, as discussed in Sec. 2.

Second, previous studies have observed the existence of a max-imum bending angle [13,15], but no comprehensive discussion ofthe reasons for the limiting behavior has been made. Also, thecited studies failed to address the impact of laser-induced imper-fections, which occur during large-bending angle experiments, onthe mechanical performance of the foam.

Finally, the cited experimental work is limited to bending anglemeasurements, and thus the existing numerical models have onlybeen validated via the bending angle [14,19]. Additionally, the pro-cess has never been simulated up to large bending angles, and nocomparison between models of different geometries has been done.

A step toward addressing the last issue was recently accom-plished by measuring the transient temperature distributions inmetal foam, using an infrared camera [21]. The obtained resultswere used to validate three numerical models with different levelsof geometrical accuracy. In this study, the efforts from Ref. [21]were extended to the analysis of the mechanical aspects of laserforming. This was done by experimentally validating the numeri-cal strain distributions using digital image correlation (DIC).Additionally, the bending mechanism was revisited by comparingmetal foam laser forming with steel sheet laser forming, as well as4-point bending. From the similarities and differences, a modifiedtemperature gradient mechanism (MTGM) has been proposed.Finally, the limiting behavior of metal foam was investigated,both experimentally and numerically, by determining the extent ofcell collapsing near the top surface, and monitoring the crack for-mation on the bottom surface. The impact of both of these imper-fections on the foam crushability and structural integrity wasinvestigated.

2 Background

2.1 Metal Foam Mechanics and Deformation. Metal foamshows fundamentally different deformation behaviors in tensionand compression. In tension, the material can undergo only asmall amount of plastic deformation and has a low strength. Incompression, on the other hand, the stress–strain curve can bedivided into three distinct stages, as shown in Fig. 1. Initially, thestress increases linearly with strain, then transitions to a large“plateau” where cells collapse, followed by an exponentialincrease after the foam is fully compressed. Due to the wide pla-teau, the area under the stress–strain curve is large, explainingwhy metal foam is an excellent energy absorber.

Metal foam bending involves a combination of tensile and com-pressive deformation. Compared to solid material, metal foam hasa very large bending stiffness S, which is defined as the resistanceof the material to bending deformation and is denoted by

S ¼ B1EI

l3(1)

where B1 is a scaling factor that is boundary and loading conditiondependent, E is Young’s modulus, I is the moment of area, and lis the beam length [3]. The reason is that the moment of area ofthe foam is more than an order of magnitude greater than themoment of area of a solid with the same net cross-sectional area(Table 1), where I is calculated via

I ¼ðs0=2

�s0=2

z2yðzÞdz (2)

where s0 is the sheet thickness and y(z) the net section width atheight z. Thus, the combination of high bending stiffness and lowtensile strength explains why mechanical bending of metal foamsis prohibitively difficult.

Unlike solid material, which is, in most cases, assumed to beincompressible, metal foam can yield due to deviatoric stresses aswell as hydrostatic stresses. Assuming isotropic behavior, the

Fig. 1 Uniaxial compressive stress–strain data (engineering)of four compression specimens (curves 1–4) made with thesame metal foam that was used in this study [22]. Thestress–strain curve can be divided into three segments: (1) alinear regime, followed by (2) a plateau where cell crushingoccurs and a large amount of energy is absorbed, followed by(3) foam densification. Due to their crushability, metal foamscan withstand much lower compressive stresses than solidmetals.

Table 1 The moment of area (about z 5 0) of a metal foam with89% porosity is 43.4 times higher than the area moment of asolid with the same cross-sectional area. As a consequence,metal foam has a much higher bending stiffness.

041004-2 / Vol. 140, APRIL 2018 Transactions of the ASME


yield surface is a closed symmetric ellipsoid with aspect ratio a,and the yield criterion involves both von Mises’ equivalent stressre and the mean stress rm [23]

F ¼ 1

1þ a=3ð Þ2r2

e þ a2r2m

� �" #1=2

� Y � 0 (3)

where Y is the uniaxial yield strength. When F< 0, the materialbehavior is elastic, and when F¼ 0, plastic deformation occurspursuant to the following flow rule:

_epij ¼

_Y

H

@F

@rij(4)

where _epij is the plastic strain rate and H is the hardening modulus

defined as

H ¼ re

r̂hr þ 1� re

r̂

� �hp (5)

where hr and hp are the tangent moduli in uniaxial and hydrostaticcompression, respectively, and r̂ is the equivalent stress that isequal to the first term in the yield criterion (Eq. (3)).

2.2 Bending Mechanism. The thermo-mechanical bendingmechanisms underlying laser forming are currently only wellunderstood for solid materials [20]. For metal foam laser forming,it has thus far always been assumed, without experimental proof,that the TGM is the governing bending mechanism.

Temperature gradient mechanism was introduced by Vollertsen[20] for sheet metal laser forming, and it governs the scenariowhere a steep temperature gradient develops across the thicknessof the workpiece. The material immediately below the laser spotheats up and tries to expand, but is restricted by the “cold” sur-rounding material. Instead of being able to expand, the materialpoint becomes plastically compressed. This phenomenon occursalong the entire scan line, making the material shorter on the topsurface and bending the workpiece toward the laser.

From a heat transfer standpoint, experiments have confirmedthat steep temperature gradients develop across metal foam duringlaser forming [21], and thus the prerequisites for TGM are met.When investigating the mechanical aspects of laser forming, how-ever, it becomes questionable whether TGM is valid for metalfoam for several reasons.

First, the “shortening” that occurs in TGM via the formation ofplastic compressive strain does not seem possible in metal foam,due to its crushability. The reason is that unless metal foam is inthe densification stage, its yield strength is less than one 60th ofthe yield strength of the solid metal it is made of, since its fragilecell walls crush when subjected to large compressive stresses.

A second argument that puts TGM in question is the low tensilestrength of metal foam. Even though TGM postulates that bendingis mainly caused by compressive strains on the top surface, sometensile deformation occurs near the bottom surface as well. Solidmetal can plastically deform in tension and thus accommodate thetensile deformation. Metal foam, on the other hand, can undergoonly a small amount of plastic tensile deformation and fracturesshortly beyond the linear elastic regime [24]. Thus, bending viaTGM should cause an immediate fracture in metal foam. Hence,the traditional TGM does not fully capture the foam response dur-ing laser forming and needs to be revisited.

2.3 Numerical Simulations. The metal foam geometry wasmodeled using two different approaches shown in Fig. 2, withdetails explained in Ref. [21]. In the first model, “equivalent”model (used in Secs. 4.1–4.3), a solid geometry with equivalentfoam properties, was used. In the second model, “Kelvin” model(used in Sec. 4.4), the foam geometry was modeled explicitly,

approximating the cavity geometry by a Kelvin-cell geometry,and assigning solid aluminum properties. The simulation was car-ried out using uncoupled thermo-mechanical analyzes, wherebythe output of the thermal analysis was used as a predefined fieldfor the mechanical analysis. While this study exclusively dis-cusses the mechanical aspects of the simulation, a detailed investi-gation of the thermal aspects may be found in Ref. [21].

In the equivalent model, the constitutive behavior was modeledusing the equations introduced in Sec. 2.1, making two majorassumptions: (1) the yield surface is symmetric, which has beenverified for a similar metal foam [23], and (2) hardening occurs inan isotropic manner. The latter assumption is not valid for largedeformations due to the fundamentally different responses ofmetal foam in tension and compression. Since the tensile strainsare small compared to the compressive strains, the induced errorsare small. In the Kelvin model, the constitutive behavior wasmodeled using von Mises’ yield criterion and the Levy–Misesflow rule.

The foam’s mechanical behavior at large bending angles wassimulated using the equivalent model. The cell crushing behaviorof the foam was modeled in an average sense, by calculating thevolumetric plastic strain rate _em from the flow rule (Eq. (4)), usingthe expression [23]

_em � _epii ¼

a2 _̂e

1þ a=3ð Þ2h i rm

r̂(6)

where _̂e is the work conjugate strain rate of r̂ that can beexpressed as _̂e ¼ _̂r=H. The volumetric strain rate can also be writ-ten as _eii ¼ _R=R, where R is the relative density and _R is the rela-tive density variation rate. Integrating this equation gives thecurrent relative density R

R ¼ R0 exp �ð

_eiidt

� �ffi R0 expð�DeiiÞ (7)

where R0 is the relative density at the previous time incrementand Deii are the logarithmic “true” strain increments in all direc-tions [25]. Equation (7) was then used to calculate the relativedensity distributions at a cross section of the foam.

The simulations were implemented in ABAQUS, and densificationcalculations were performed in MATLAB. Multiscan simulationswere performed on the Stampede supercomputer provided byExtreme Science and Engineering Discovery Environment [26].Quadratic elements C3D20 were used for the equivalent model,and linear tetrahedral elements C3D4 were used for the Kelvinmodel. A y-symmetry boundary condition was used that set dis-placements in the y-direction (U2) and rotations about the x- and

Fig. 2 Two approaches were used to model metal foam. In thefirst (equivalent) model, shown in (a), a solid geometry wasused and equivalent foam properties were assigned. In the sec-ond (Kelvin) model, shown in (b), the foam geometry wasexplicitly modeled, and solid aluminum properties wereassigned. The cavity geometry was approximated by a Kelvin-cell geometry.

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z-axes (UR1, UR3) equal to zero. Furthermore, the displacementsin x and z (U1, U3) were restricted at two vertically aligned pointson the symmetry surface. Rotations about the y-axis (UR2) werenot restricted.

The thermal properties were used from Ref. [21]. Thetemperature-dependent thermal expansion coefficient ath(T) of thefoam was assumed to be identical to that of the correspondingsolid and was extracted from Ref. [27]. Temperature-dependentYoung’s modulus E(T) was obtained for solid AlSi11 [28] andconverted to foam using relations given in Ref. [29]. The elasticand plastic Poisson’s ratios t and tp were obtained from Refs. [23]and [29], respectively. The flow stress rf was obtained from Ref.[22], and the temperature dependence was adapted from solid alu-minum data [27]. The compressive yield stress ratio r/pc was esti-mated based on Ref. [23].

Steel sheet forming simulations were performed using themodel in Ref. [11]. 4-point bending simulations were based on U-bend simulations developed by Brandal and Yao [30]. The loadingblock and supports were defined as rigid shells, and a general con-tact interaction, with zero friction, was used. Loading was exertedby vertically displacing the loading block.

3 Experimental Procedures

Closed-cell Al foam was used, with 7 wt % silicon, a volumefraction of 11.2%, and a density of 279 kg/m3. The foam was man-ufactured employing a melt-foaming method that used TiH2 as afoaming agent and calcium to increase the viscosity of the liquidaluminum. Slitting cutters and end mill tools were used to cut testspecimens to a length, width, and thickness of 100 mm, 35 mm,and 10 mm, respectively.

Laser forming experiments were performed using a continuous-wave Nd:YAG laser with a wavelength of 1064 nm. All theexperiments were performed using a spot size of 12 mm, and thespecimens were positioned below the focal plane to ensure thatthe laser intensity does not increase inside cavities (Fig. 3(a)). Thespecimens were clamped at one end using rubber pads to providethermal insulation. Nitrogen was used as a protective gas to avoidoxidation, and in between successive scans, the specimen wasallowed to cool to room temperature to prevent heat accumulationeffects. The bending angle was determined by measuring the ver-tical deflection with a dial indicator.

The specimens were scanned underneath the laser using a sixdegrees-of-freedom St€aubli RX-130 robot. Two representativeprocessing conditions were contrasted with (power, scan speed) of(90 W, 5 mm/s) and (180 W, 10 mm/s), respectively. The line

energy LE¼P/v was kept constant, since constant LE experi-ments reveal several aspects of the physical behavior of metalfoam and allow for a meaningful comparison with numerical sim-ulations as has been shown in Ref. [31].

Digital image correlation was used to determine the strain dis-tribution on the bottom specimen surface in between consecutivelaser scans (Fig. 3(b)). The DIC experiments were performed byspray-painting the bottom specimen surfaces white and subse-quently applying a black speckle pattern. A digital camera with aresolution of 2448� 2048 pixels was used to take images of thespeckle pattern after each laser scan. To achieve the best resolu-tion of about 100 lm, it was ensured that the speckle sizes corre-sponded to 3–5 pixels on the digital camera [32,33]. Thecommercial DIC software VIC-2D from correlated solutions wasused to calculate the Lagrangian strain fields from the digitalimages, which were subsequently converted to the logarithmic“true” strain fields. All the strains were computed in the tensiledirection (eyy). Several calibration tests were performed to ensurethat the results were independent of variables such as the subset,stepsize, seed placement, and incremental correlation. An exam-ple of a processed image is shown in Fig. 3(b), representing astrain distribution (eyy) at a bending angle of 45 deg.

To avoid strains induced by out-of-plane rotations [34], thestrain was only extracted on the clamped half of the specimenabove the bending axis (Fig. 3(b)). To minimize the impact oflocal inhomogeneities, the data were averaged over the entirespecimen width and a distance of 5 mm from the bending axis.Standard errors were calculated over all individual pixels withinthat area.

The laser forming strain distributions were compared withstrain distributions in 4-point bending experiments. 4-point bend-ing tests were performed pursuant to a combination of ASTMstandards D7249 and C1341. The data were averaged over a rec-tangle similar to laser forming, and was also corrected for strainscaused by out-of-plane displacements [34].

4 Results and Discussion

4.1 Bending Mechanism. In Sec. 2, it was predicted thatmetal foam is unable to develop the compressive strains central tothe traditional TGM, due to its crushability and low compressivestrength. Laser forming experiments revealed two aspects of thefoam behavior that spoke in favor of this prediction. First, no mat-ter how low within the processing window the laser power waschosen, localized melting of thin cell walls was unavoidable. As

Fig. 3 (a) Experimental setup. The laser was scanned in x-direction. The bottom specimen sur-face was spray-painted white with a black speckle pattern. Digital images were taken in betweenconsecutive laser scans. (b) Example strain distribution (eyy, Lagrangian strain) on the bottomsurface of a laser formed specimen at a bending angle of 45 deg. The strain was only extractedon the clamped half of the specimen above the bending axis to avoid effects related to out-of-plane rotations [34].



shown in Fig. 4, melting initiated at the first laser scan and pro-gressed with each successive laser scan. Thin cell walls startedmelting from the top surface, forming u-shaped trenches thatdeepened until either the entire cell wall was melted away, or thecell wall thickness increased. If metal foam bending indeedoccurred due to plastic compressive strains, melting would drasti-cally impede bending, because it reduces the amount of compress-ible material. Experiments have shown the contrary, however,implying that plastic compressive strains cannot be the majorcause of bending. The second aspect is that cell wall bendingoccurred close to the bending axis (Fig. 5), clearly indicating thatthe cell walls are unable to withstand high compressive stresses.

An alternative explanation is that metal foam bending is mainlycaused by tensile stretching near the bottom surface instead ofcompressive deformation on the top. To investigate this hypothe-sis, the tensile strain eyy (determined via DIC) was compared forlaser forming and 4-point bending (Fig. 6). eyy is the strain result-ing from a combination of cell collapsing and cell wall deforma-tion. In laser forming, eyy indeed grew substantially larger than in4-point bending, due to heat-induced softening. However, whilethe maximum eyy in laser forming was around four times greaterthan in 4-point bending, the maximum bending angle was greaterby a factor of 7. Therefore, eyy did not grow proportionally to thebending angle; hence, tensile stretching cannot possibly accountfor all of the bending deformation.

Numerical results confirmed that tensile stretching is not thedriving force of the bending deformation. Figure 7(d) shows theratio of the compressive top surface strain to the tensile bottom

surface strain for metal foam laser forming, steel sheet laser form-ing, and 4-point bending (models shown in Figs. 7(a)–7(c)). Theequivalent model was used for the simulation of metal foam,which was experimentally validated in Refs. [21] and [35]. In allcases, the strains were extracted at the bending axis. For foam andsteel laser forming, the ratio was averaged over 10 scans and twotypical conditions [foam—(90 W, 5 mm/s; 180 W, 10 mm/s),steel—(400 W, 25 mm/s; 800 W, 50 mm/s)], to obtain a most

Fig. 4 Section of the top surface of the specimen on the laser scan line (a) before laser form-ing, (b) after 1 scan, and (c) after 10 scans at 90 W and 5 mm/s. Even at this low power, localizedmelting of thin cell walls occurred, marked in white. Melting started after the first scan and pro-gressed with each consecutive scan, forming u-shaped trenches in the cell walls and reducingthe amount of compressible material. Melting stopped once the thickness of the remaining cellwall sections increased.

Fig. 5 Cross section of a foam specimen after five laser scansat 180 W and 10 mm/s. The laser was scanned into the page.The white arrows show where cell walls were bent during laserforming, indicating that the foam cannot withstand high com-pressive stresses.



representative value. For 4-point bending, the ratio was averagedover all the bending angles until the experimental failure angle.The results in Fig. 7(d) clearly show that; whereas compressionand tension contributed equally in 4-point bending, compressivedeformation was the major cause of bending in laser forming.Interestingly, metal foam and steel sheet laser forming nearlyyielded the same ratio, implying that the compressive“shortening” on the top surface should be identical in both metalfoam and steel.

Numerical relative density distributions at a cross section, cal-culated after a single laser scan at 180 W and 10 mm/s (Fig. 8),allow a similar conclusion to be drawn. The baseline relative den-sity is 0.112, and a relative density of 1 indicates complete densifi-cation. The densified (compressed) region stretched over the top80% of the foam, leaving only a small expanded (tensile) regionon the bottom. Therefore, laser forming seems to shift down the

Fig. 6 Comparison of the tensile strain (eyy) on the bottom sur-face (determined via DIC) in 4-point bending and laser forming(at 180 W and 10 mm/s). eyy is the strain resulting from a combi-nation of cell collapsing and cell wall deformation. Standarderrors are shown, calculated over all the pixels of the averagedareas (see Sec. 3). In 4-point bending, the strain increasedexponentially before a catastrophic failure, whereas in laserforming there was a stable linear strain growth with increasingbending angle. Laser forming yielded a larger maximum eyy, dueto heat-induced softening, yet eyy did not grow proportionally tothe bending angle, and hence tensile stretching cannot be themain cause of bending.

Fig. 7 Strain distributions (eyy) in (a) metal foam during 4-point bending, (b), metal foam during laser forming, and (c) a steelsheet during laser forming. (d) Ratio of top surface (compressive) strain over bottom surface (tensile) strain at the bending axis,calculated for all three simulations. In 4-point bending, bending was equally caused by compressive and tensile strains. In laserforming, the large ratios indicate that compressive deformation was the main cause for bending.

Fig. 8 Relative density distribution after a single scan at 180 Wand 10 mm/s. The baseline relative density is 0.112, and a rela-tive density of 1 indicates complete densification. Densificationoccurred throughout the top 80% of the foam, leaving only asmall tensile region on the bottom surface. Therefore, laserforming shifts the neutral axis downward, and deformation isdominated by compressive shortening.



neutral axis and limits the amount of tensile deformation occur-ring, unlike in 4-point bending (Fig. 7(a)), where the neutral axislies midway through the thickness (assuming small bendingangles).

The findings in Figs. 4–8 allow only one explanation: cell wallbending and cell crushing in metal foam laser forming, observedin Fig. 5, is equivalent to plastic compressive strains in steel sheetlaser forming.

Using that insight, the bending mechanism of metal foam canbe revisited. Metal foam develops steep temperature gradientsacross its thickness and thus meets the thermal prerequisites ofTGM [21]. Similar to solid metal, bending is mainly achieved viacompressive deformation, and tensile deformation near the bottomsurface only occurs to accommodate the “shortening” on the topsurface. The reason for its “shortening” near the top surface, how-ever, differs from conventional TGM. Unlike solid sheet metal,which develops plastic compressive strains, metal foam shortensdue to cell wall bending and cell collapsing. Moreover, metalfoam does not immediately undergo a tensile fracture on the bot-tom surface due to heat-induced softening, as well as a downwardshift in the neutral axis that limits the amount of tensile deforma-tion. Since metal foam still meets most of the requirements set byTGM, this revised bending mechanism may be called MTGM.

4.2 Bending Limit. For laser forming of solid sheet metal,several studies reported that the bending increment decreases withincreasing laser scans [36,37]. In laser forming of metal foam, asimilar limiting behavior was observed at two different processingconditions, (high—180 W; 10 mm/s) and (low—90 W; 5 mm/s),shown in Fig. 9. In the experiment, as well as the simulation, therate of change of both the bending angles and tensile strainsdecreased with an increasing number of scans and eventuallyapproached zero, indicating that there is a maximum achievablebending angle.

In solid metals, the limiting behavior can be attributed toincreases in the sheet thickness, strain hardening on the bottomsurface, and variations in the laser absorption due to coatingremoval and heat-induced surface discoloration [36,37]. In metalfoam, on the other hand, thickening does not occur, and neitherdoes strain hardening on the top surface, since the flow stressremains constant until densification. Tensile strain hardening onthe bottom surface only occurs to a small extent because metalfoam can only undergo limited amounts of plastic tensile defor-mation. Variations in the laser absorption do occur in metalfoams, since cavities crush and rotate with increasing bendingangle, but they are not significant enough to be the major cause ofthe bending limit. Similarly, melting ceases after a few laser scansand cannot be responsible for the limit either.

Therefore, the bending limit must be caused by a phenomenonthat does not occur in sheet metal laser forming, which is cellcrushing and the subsequent densification. Densification increasesthe amount of material away from the central axis of the foam,and hence increases its moment of area, rendering the foam stiffer.Simultaneously, both Young’s modulus [29] and the flow stress[1] of the foam increase exponentially with its relative density, asshown in

Ef

Es¼

1� qf =qs

� �2

1þ 2� 3tsð Þqf =qs

� � (8)

rf

rs� 0:3 /

qf

qs

� �32

þ 1� /ð Þqf

qs

(9)

where Ef ; rf ; and qf are the Young’s modulus, yield strength,and density of the foam, respectively, and Es; rs; and qs are thecorresponding solid properties. ts is the Poisson’s ratio of thesolid, and / is the percentage of solid material at cell intersec-tions, which was assumed to linearly increase with foam density.

These equations show that densification stiffens the foam evenfurther and reduces the amount of plastic deformation. Densifica-tion also increases the thermal conductivity, as is shown by [3]

qf

qs

� �1:8

<kf

ks<

qf

qs

� �1:65

(10)

where kf and ks represent the foam and solid thermal conductiv-ities, respectively. As a consequence, the heat diffusion awayfrom the top surface increases, thereby reducing the amount ofthermal expansion and plastic compressive deformation. Hence,densification is the main reason for the bending limit in metalfoam laser forming.

From the results in Fig. 9, it is apparent that (180 W; 10 mm/s)yielded a higher maximum bending angle than (90 W; 5 mm/s).The reason is that, for an increased power and speed (atLE¼ const.), heat has less time to dissipate, thus increasing thetemperature rise, thermal expansion, and plastic compressive

Fig. 9 (a) Experimental and (b) numerical bending angles andtensile strain data (eyy) at large bending angles. “High” refers tothe condition 180 W 10 mm/s, “low” refers to 90 W 5 mm/s.Standard errors are shown for the strain results, calculatedover all the pixels of the averaged area (see Sec. 3). In both theexperiment and the simulation, the bending angle and strainplots leveled off at a large number of scans. In the numericalsimulation, the limit at 180 W and 10 mm/s was reached at alarger bending angle, since the model did not consider melting,as well as changes in the moment of area, laser absorption andthermal conductivity with increasing densification.



“shortening” close to the top surface [21,31]. Hence, more densifi-cation is required to stop the plastic compression from occurring.

Numerical results confirm the above reasoning. Withoutaccounting for densification effects, the bending and strain incre-ment only slightly decreased with increasing laser scans inFig. 10, whereby the decrease was caused by tensile hardening onthe bottom surface. When calculating the density as a field vari-able using q ¼ q0e�ðeiiÞ (from Eq. (7)) and incorporating density-

dependent Young’s moduli and flow stresses from Eqs. (8) and(9), both the bending and strain increment decreased much morequickly, and the resulting corrected results agreed better with theexperimental trends. Therefore, simulations confirmed thatdensification-induced changes in the material properties are themajor cause of the bending limit.

While the numerical results agreed with the experimentalresults at 90 W and 5 mm/s, they overestimated the bending angleat 180 W and 10 mm/s, and simultaneously underestimated thetensile strain on the bottom surface. Several approximations in themodel led to this error that only became significant at large bend-ing angles and higher powers/speeds. First, melting was neglected,which relaxes stresses in the material and thus reduces the amountof bending. Second, the model failed to account for changes in themoment of area, owing to its solid geometry. Finally, since it is anuncoupled analysis, the model did not account for changes in thelaser absorption, as well as densification-induced changes in thethermal conductivity, k, whose impact can become significant atlarge bending angles. This problem could be mediated by using acoupled thermo-mechanical analysis, but doing so currently ren-ders the simulation too CPU-intensive to study laser forming atlarge bending angles.

4.3 Impact of Laser Forming on Metal Foam Performance.Bending deformation in metal foam laser forming is achieved byintroducing irreversible plastic deformation, which, as was shownpreviously, manifests itself mainly through cell crushing and sometensile deformation. This plastic deformation may affect the per-formance of metal foam as a crash absorber and structural member.In this section, it is demonstrated that the impact of laser-inducedimperfections is negligible until large bending angles are reached.

A first type of laser-induced imperfections is crack formationnear the bottom surface. Some thin cell walls, which have

Fig. 10 Comparison of numerical data before and after correct-ing for density dependence. Without correction, both the bend-ing angle and strain increment decreased only slightly withincreasing scans, induced by tensile strain hardening on thebottom surface. After incorporating density-dependent varia-bles, a distinct limiting behavior could be observed.

Fig. 11 Bottom surface of a specimen scanned at 90 W and 5 mm/s at bending angles of (a)1 deg, (b) 11.6 deg, and (c) 16.3 deg. Due to some naturally occurring stress concentrations,microcracks already occurred at low bending angles of 1 deg. As the bending angle becamelarger, those microcracks increased in size, and new microcracks formed ((b) and (c)). However,the microcracks remained isolated from each other at this stage and thus hardly affected thestructural integrity of the foam.



naturally occurring stress concentrations, while also being favor-ably aligned with the bending direction, can form microcracksalready at low bending angles of 1 deg (Fig. 11(a)). More micro-cracks form with increasing bending angles (Figs. 11(b) and11(c)), but they are small in size and remain isolated from eachother. Once the bending angle approaches 45 deg, the cracks growand start coalescing, forming a more visible crack (Fig. 12(a)).

The impact of those micro and macrocracks on the structuralstrength of metal foam can be determined by analyzing its fracturebehavior. Sugimura et al. [38] and Mccullough et al. [39] showedthat metal foam does not fracture in a “clean” manner. Instead, thecellular structure makes crack propagation rather tortuous, andmajor cracks in the foam are always preceded, and accompaniedby, small side cracks. Therefore, even in the presence of micro-cracks, metal foam is still far away from complete failure, unlikea typical brittle material, in which the smallest crack can initiate acatastrophic failure.

Further insight can be obtained by evaluating the J-value,which is defined as the decrease in potential energy due to thegrowth of an incremental crack length, Da [40]. Figure 13 showsthe J-integral resistance curve obtained by Sugimura et al. for asimilar foam [38]; comparable results were obtained by Mccul-lough et al. [39]. The exact crack length, up to which the J-integral resistance curve is reliable, depends on the precise speci-men geometry, particularly the distance that is available for crackgrowth as per ASTM E813-89. Regardless of the geometry, how-ever, the J-curve in Fig. 13 shows a clear tendency to increasewith crack length, unlike for brittle materials, where the J-curvedescribes a horizontal line beyond an unstable crack length [41].Therefore, the microcracks observed in Fig. 11 hardly reduce thecrack resistance, and even large cracks do not entirely remove thestructural integrity of the foam.

While the aforementioned fracture analysis applies to mechani-cally fractured aluminum foam, it can be shown that the fracturebehavior of laser-formed metal foam is even less of concern. Fig-ure 14 shows a comparison between the fracture surfaces in 4-point bending and laser forming. Whereas the fracture surface in4-point bending contained a mix of rough ductile dimples andclean brittle surfaces with sharp edges, the material on the laserformed fracture surface appears to have been pulled toward theright side, and thus has undergone a substantial amount of plasticdeformation before fracture. Hence, the fracture in laser formingwas much more ductile, due to a heat-induced reduction of theflow stress, which explains why laser forming yielded much largertensile strains eyy in Fig. 6. This result further emphasizes that thestructural integrity of laser formed metal foam, even more so thanuntreated foam (in Fig. 13), is hardly affected by microcracksuntil large bending angles are reached.

The second type of imperfection introduced by laser forming iscell crushing, first discussed in Fig. 5, and now shown intensifiedat a large bending angle in Fig. 12(b). In order to determine theimpact of cell crushing on the foam crushability, the changes in

the relative density at large bending angles were analyzed, shownin Fig. 15. Even at a bending angle of 45 deg, the maximum rela-tive density achieved was 0.271, which is far away from completedensification of R¼ 1. Figure 15 further shows that the com-pressed area is highly localized around the bending axis, which isalso the case in the experiment (Fig. 12(b)). Therefore, while cellcrushing does to an extent reduce the compressibility of metalfoam, it occurs in a highly localized manner, and to a degree thatdoes not severely hamper the performance of metal foam as acrash absorber.

4.4 Alternative Numerical Models. So far, the numericalsimulations were performed using a solid geometry model, whichwas found to induce errors due to its inaccurate laser absorption[21], as well as its inability to model changes in the moment of

Fig. 12 (a) Foam specimen at a bending angle of 45 deg, after being laser formed at 180 W and10 mm/s, with a cross section shown in (b). At large bending angles, the isolated cracks on thebottom surface grew larger and started to coalesce, shown in (a). In close proximity to the bend-ing axis, cells at the top surface were crushed significantly, shown in (b), but the foam was stillfar away from complete densification.

Fig. 13 Resistance curves determined using the J-integralmethod, ASTM 813-89, for a closed-cell foam with a very similarcomposition and porosity [38]. The white and black data pointsrepresent two test specimens. Unlike in a brittle material, wherethe JR-value would be horizontal beyond an unstable cracklength, it keeps on increasing in metal foam, indicating that thefoam fracture toughness is maintained even as the crack growslarger. Therefore, microcracks do not lead to an unstable frac-ture, and even larger cracks do not completely remove thestructural integrity of the foam.



area with increasing bending angle. These issues can be remediedby explicitly modeling the foam geometry.

Figure 16 shows the bending angle predictions of a Kelvinmodel for 8 laser scans, in comparison with the equivalent modelpredictions and the experimental data. The high and low condi-tions again refer to (180 W; 10 mm/s) and (90 W; 5 mm/s), respec-tively. On the one hand, the Kelvin model yielded results thatmore closely matched with the experimental values, due to itsimproved geometrical accuracy. On the other hand, the model stilloverestimated the bending angle at 180 W and 10 mm/s, since itdid not account for melting, as well as changes in the laser absorp-tion with increasing bending angle (due to using an uncoupledanalysis). Also, the model geometry was too simplified to accu-rately predict crack initiation sites or cell wall bending.

A more accurate geometry can be obtained by “randomly” dis-persing unit cells of different sizes throughout the model geome-try, as was done by De Giorgi et al. [42] and Roohi et al. [19] forlow- and high-density foams, respectively. However, the“random” dispersion of the cells still follows systematic algo-rithms, and the unit cell geometry remains highly simplified.

In order to obtain genuine randomness, a micro-CT based“voxel” model could be used. From a thermal standpoint, it hasalready been shown that the voxel model can yield good results[21]. Figure 17 demonstrates that the voxel model can also be

used to model the mechanical response of the foam, showing thethermal strain distribution during a laser scan at 90 W and 5 mm/s.Due to its geometrical accuracy, particularly its capacity to modeleven thin cell walls, the voxel model could potentially be used topredict early-stage cracking and cell wall bending.

Despite these advantages, it is often not worth going to the greatlengths of using a voxel model, since the modeling techniquecomes along with several disadvantages. First, each voxel modelrepresents only one particular metal foam specimen, and itbecomes necessary to generate a new voxel model for each speci-men if a high level of accuracy is desired. Second, due to itsextremely large number of elements, the CPU intensity increasesdramatically, and the specimen size that can be modeled is lim-ited. Due to these significant drawbacks, the equivalent and Kel-vin models are in most cases adequate to model the metal foamresponse during laser forming, despite their limited geometricalaccuracy.

Fig. 14 Fracture surfaces in (a) mechanically fractured and (b) laser formed metal foam specimens. The fracture surface in (a)consists of a mix of dimples and clean surfaces with sharp edges, indicating a mix of ductile and brittle fracture. The materialon the surface of (b), on the other hand, appears to have been stretched severely, indicating a more ductile fracture.

Fig. 15 Relative density distribution at a bending angle of45 deg. The baseline relative density is 0.112, and a relative den-sity of 1 indicates complete densification. The densified regionwas highly localized around the bending axis. Over a small areaclose to the top surface, the relative density increased by a fac-tor of 2.5, which is still far away from complete densification.Foam expansion occurred close to the bottom surface, coincid-ing with the region where cracks appeared in Fig. 12(a).

Fig. 16 Bending angles predicted by the Kelvin model in com-parison with the predictions of the equivalent model and theexperimental values. Standard errors are shown for the experi-mental data. Due to its superior geometrical accuracy, the Kel-vin model yielded results that were closer to the experimentalresults. Nevertheless, the model still overestimated the bendingangle, particularly at the high condition (180 W, 10 mm/s), sinceit did not account for melting, as well as changes in the laserabsorption (due to using an uncoupled modeling approach).



5 Conclusions

In this study, the mechanical response of metal foam duringlaser forming was investigated. Metal foam was found to undergocompressive shortening via cell wall bending and cell crushingnear the irradiated surface, as opposed to compressive plasticstrains that occur in laser forming of solid sheet metal. Based onthis deviation from the traditional TGM, a MTGM was proposed.From bending angle and strain measurements (determined viaDIC), it was found that the achievable bending angle is limited,which can be attributed mostly to densification-induced changesin the thermal conductivity, material stiffness, flow stress, andmoment of area. At bending angles around 45 deg, crack forma-tion and cell crushing were observed. It was shown, however, thatthese cracks do not negatively impact the structural integrity ofthe foam, and the extent of the cell crushing has a minor impacton the foam crushability, since the foam remains far away fromfull densification and the densification is localized.

Numerically, the mechanical response of metal foam was mod-eled using an equivalent model, which captured the experimentaltrends despite using a highly simplified geometry. Alternativemodeling approaches using different explicit foam geometrieswere discussed.

Acknowledgment

The authors are grateful to Professor J. W. Kysar and Bob Starkfor providing us with the DIC equipment and software.

Funding Data

� National Science Foundation (Grant No. CMMI-1725980).� Extreme Science and Engineering Discovery Environment

(XSEDE) Stampede through allocation TG-DDM160002,which is supported by National Science Foundation (GrantNo. ACI-1548562).

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